Spherical Harmonic Gravity Model Research Articles

Dynamic Model Fidelity Effects on Covariance Based Track Association

This paper examines the performance of covariance-based track association with varying levels of dynamic model fidelity. To minimize the dynamic nonlinearities, the mean modified equinoctial (MME) elements are used. Ground-based radar measurements are generated by propagating the osculating orbits of 100 space objects in low Earth orbit using a 70×70 spherical harmonic gravity model (JGM-3). Atmospheric drag, solar radiation, and third-body gravity are not considered. To associate two tracks, the two sets of states and covariances are propagated to a common time. The Mahalanobis distance between the two tracks is then used to determine whether they associate. A higher-fidelity dynamic model for state propagation improves association performance, while the fidelity of the dynamic model for covariance propagation does not have a significant impact. The MME-based track association demonstrates excellent performance when incorporating J2, J22, J4, and J6 in the MME state propagation as well as the MME covariance propagation with J2. It achieves an accuracy of 99.92%, a true positive sensitivity of 92.43%, a false negative sensitivity of 7.57%, a false positive sensitivity of 0%, and a true negative sensitivity of 100% when associating tracks separated by 10 days.

Analytical Radial Adaptive Method for Spherical Harmonic Gravity Models

Analytical Radial Adaptive Method for Spherical Harmonic Gravity Models

Residual terrain modelling (RTM) in terms of the cap-modified spectral technique: RTM from a new perspective

We present two novel approaches to residual terrain modelling (RTM), one of which is practical and the other rather theoretical. The first provides a solution to the harmonic correction issue and the high-frequency error of the spectral filter problem. As its key feature, cap-modified spectral gravity forward modelling is applied to deliver near-zone gravity effects induced by the reference (smooth) topography. Thanks to its spectral nature, the sought gravity can be evaluated at the problematic points inside the smooth topography by regularized downward continuation (solution for the harmonic correction problem), and, at the same time, it is band-limited in spherical harmonics (solution for the high-frequency error). A validation over two mountainous areas, Switzerland and Slovakia, reveals that this technique is at least comparable with two other common RTM variants (RMS agreement up to 0.1 mGal). Finally, we formulate the theoretical RTM concept, showing that the harmonic correction issue and the spectral filter problem are caused by filtering of the mass model in the topography domain. When properly filtering gravity effects in the gravity domain, that is, avoiding the concept of the reference topography, both problems disappear. Limitations of both RTM approaches include the possible divergence effect of spherical harmonic series on the Earth’s surface and a conceptual inconsistency between two involved types of spherical harmonic coefficients. Applications of this study could be found in the development of combined gravity models (the fill-in procedure) or in the spectral enhancement of spherical harmonic gravity models.

Efficient Computation of Optimal Low Thrust Gravity Perturbed Orbit Transfers

We have developed a new method for solving low-thrust fuel-optimal orbit transfer problems in the vicinity of a large body (planet or asteroid), considering a high-fidelity spherical harmonic gravity model. The algorithm is formulated via the indirect optimization method, leading to a two-point boundary value problem (TPBVP). We make use of a hyperbolic tangent smoothing law for performing continuation on the thrust magnitude to reduce the sharpness of the control switches in early iterations and thus promote convergence. The TPBVP is solved using the method of particular solutions (MPS) shooting method and Picard-Chebyshev numerical integration. Application of Picard-Chebyshev integration affords an avenue for increased efficiency that is not available with step-by-step integrators. We demonstrate that computing the particular solutions with only a low-fidelity force model greatly increases the efficiency of the algorithm while ultimately achieving near machine precision accuracy. A salient feature of the MPS is that it is parallelizable, and thus further speedups are available. It is also shown that, for near-Earth orbits and over a small number of en-route revolutions around the Earth, only the zonal perturbation terms are required in the costate equations to obtain a solution that is accurate to machine precision and optimal to engineering precision. The proposed framework can be used for trajectory design around small asteroids and also for orbit debris rendezvous and removal tasks.

Divergence-free spherical harmonic gravity field modelling based on the Runge–Krarup theorem: a case study for the Moon

Recent numerical studies on external gravity field modelling show that external spherical harmonic series may diverge near or on planetary surfaces. This paper investigates an alternative solution that is still based on external spherical harmonic series, but capable of avoiding the divergence effect. The approach relies on the Runge–Krarup theorem and the iterative downward continuation. In theory, Runge–Krarup-type solutions are able to approximate the true potential within the entire space external to the masses with an arbitrary $$\varepsilon $$ -accuracy, $$\varepsilon >0$$ . Using gravity implied by the lunar topography, we show numerically that this technique avoids indeed the divergence effect, at least at the studied 5 arc-min resolution. In the context of the iterative scheme, we show that a function expressed as a truncated solid spherical harmonic expansion on a general star-shaped surface possesses an infinite surface spherical harmonic spectrum after it is mapped onto a sphere. We also study the convergence of the gradient approach, which is a technique for efficient grid-wise synthesis on irregular surfaces. We show that the resulting Taylor series may converge slowly when analytically upward continuing from points inside the masses. The continuation from the mass-free space should therefore be preferred. As an underlying topic of the paper, spherical harmonic coefficients from spectral gravity forward modelling and their Runge–Krarup counterpart are numerically studied. Regarding their different nature, we formulate some research topics that might contribute to a deeper understanding of the current methodologies used to develop combined high-degree spherical harmonic gravity models.

An Online Gravity Modeling Method Applied for High Precision Free-INS.

For real-time solution of inertial navigation system (INS), the high-degree spherical harmonic gravity model (SHM) is not applicable because of its time and space complexity, in which traditional normal gravity model (NGM) has been the dominant technique for gravity compensation. In this paper, a two-dimensional second-order polynomial model is derived from SHM according to the approximate linear characteristic of regional disturbing potential. Firstly, deflections of vertical (DOVs) on dense grids are calculated with SHM in an external computer. And then, the polynomial coefficients are obtained using these DOVs. To achieve global navigation, the coefficients and applicable region of polynomial model are both updated synchronously in above computer. Compared with high-degree SHM, the polynomial model takes less storage and computational time at the expense of minor precision. Meanwhile, the model is more accurate than NGM. Finally, numerical test and INS experiment show that the proposed method outperforms traditional gravity models applied for high precision free-INS.

Application of the spherical harmonic gravity model in high precision inertial navigation systems

The spherical harmonic gravity model (SHM) may, in general, be considered as a suitable alternative to the normal gravity model (NGM), because it represents the Earth’s gravitational field more accurately. However, the high-resolution SHM has never been used in current inertial navigation systems (INSs) due to its extremely complex expression. In this paper, the feasibility and accuracy of a truncated SHM are discussed for application in a real-time free-INS with a precision demand better than 0.8 nm h−1. In particular, the time and space complexity are analyzed mathematically to verify the feasibility of the SHM. Also, a test on a typical navigation computer shows a storable range of cut-off degrees. To further evaluate the appropriate degree and accuracy of the truncated SHM, analyses of covariance and truncation error are proposed. Finally, a SHM of degree 12 is demonstrated to be the appropriate model for routine INSs in the precision range of 0.4–0.75 nm h−1. Flight simulations and road tests show its outstanding performance over the traditional NGM.

New Solutions for the Perturbed Lambert Problem Using Regularization and Picard Iteration

A new approach for solving two-point boundary value problems and initial value problems using the Kustaanheimo–Stiefel transformation and Modified Chebyshev–Picard iteration is presented. The first contribution is the development of an analytical solution to the elliptic Keplerian Lambert problem based on Kustaanheimo–Stiefel regularization. This transforms the nonlinear three-dimensional orbit equations of motion into four linear oscillators. The second contribution solves the elliptic Keplerian two-point boundary value problem and initial value problem using the Kustaanheimo–Stiefel transformation and Picard iteration. The Picard sequence of trajectories represents a contraction mapping that converges to a unique solution over a finite domain. Solving the Keplerian two-point boundary value problem in Kustaanheimo–Stiefel variables increases the Picard domain of convergence from about one-third of an orbit (Cartesian variables) to over 95% of an orbit (Kustaanheimo–Stiefel variables). These increases in the domain of Picard iteration convergence are independent of eccentricity. The third contribution solves the general spherical harmonic gravity perturbed elliptic two-point boundary value problem using the Kustaanheimo–Stiefel transformation and Picard iteration, and it does not require a Newton-like shooting method for fractional orbit transfers. For multiple revolution transfers, however, a shooting method can make use of the Modified Chebyshev–Picard iteration/Kustaanheimo–Stiefel/initial value problem and the Method of Particular Solutions to obtain solutions given a Keplerian Lambert solution as the starting iterative. The Kustaanheimo–Stiefel perturbed solution is illustrated using a (40,40) degree and order spherical harmonic gravity model. A general three-dimensional recipe is introduced for solving the perturbed Lambert Problem via Modified Chebyshev–Picard iteration without a Newton-like shooting method for the fractional orbit case. The increase in the domain of convergence of the Kustaanheimo–Stiefel transformed, perturbed Lambert problem via Modified Chebyshev–Picard iteration versus the Cartesian Modified Chebyshev–Picard iteration Lambert solution is analogous to the results for the Keplerian case. The three-dimensional two-impulse perturbed Lambert problem is efficiently convergent up to about 85% of the Keplerian orbit period with a (40,40) spherical harmonic gravity model. The efficiency of the current two-point boundary value problem solver is compared with MATLAB’s fsolve, where Runge–Kutta–Nystrom 12(10) and Gauss–Jackson are the integrators.

Assessment of systematic errors in the surface gravity anomalies over North America using the GRACE gravity model

SUMMARY The surface gravity data collected via traditional techniques such as ground-based, shipboard and airborne gravimetry describe precisely the local gravity field, but they are often biased by systematic errors. On the other hand, the spherical harmonic gravity models determined from satellite missions, in particular, recent models from CHAMP and GRACE, homogenously and accurately describe the low-degree components of the Earth’s gravity field. However, they are subject to large omission errors. The surface and satellite gravity data are therefore complementary in terms of spectral composition. In this paper, we aim to assess the systematic errors of low spherical harmonic degrees in the surface gravity anomalies over North America using a GRACE gravity model. A prerequisite is the extraction of the low-degree components from the surface data to make them compatible with GRACE data. Three types of methods are tested using synthetic data: low-pass filtering, the inverse Stokes integral, and spherical harmonic analysis. The results demonstrate that the spherical harmonic analysis works best. Eighty-five per cent of difference between the synthetic gravity anomalies generated from EGM96 and GGM02S from degrees 2 to 90 can be modelled for a region covering North America and neighbouring areas. Assuming EGM96 is developed solely from the surface gravity data with the same accuracy and GGM02S errorless, one way to understand the 85 per cent difference is that it represents the systematic error from the region of study, while the remaining 15 per cent originates from the data outside of the region. To estimate systematic errors in the surface gravity data, Helmert gravity anomalies are generated from both surface and GRACE data on the geoid. Their differences are expanded into surface spherical harmonics. The results show that the systematic errors for degrees 2 to 90 range from about −6 to 13 mGal with a RMS value of 1.4 mGal over North America. A few significant data gaps can be identified from the resulting error map. The errors over oceans appear to be related to the sea surface topography. These systematic errors must be taken into consideration when the surface gravity data are used to validate future satellite gravity missions.

Stress and Flexural Modeling of the Martian Lithospheric Response to Alba Patera

The martian volcano Alba Patera is surrounded by a system of graben which appears to be the result of an extensional stress applied to the region after the construction of the volcano. These faults provide a map of the stress field that created them. We used this information, as well as the recent Goddard spherical harmonic gravity model (GMM-1) and a spherical harmonic representation of the Mars Digital Elevation Model, to constrain models of: (1) the lithospheric flexure under the load of the volcano, (2) the gravitational anomaly due to the volcano, and (3) the magnitude of the extensional stress in the region. We found that elastic lithospheric thicknesses between 10 and 50 km resulted in statistically acceptable fits to the gravity and topography data. We also found that a 1.5–2 kilobar extensional regional stress in conjunction with the axisymmetric stress field of the volcanic load produced a stress field that is consistent with the observed trends of the graben.

Venus Spherical Harmonic Gravity Model to Degree and Order 60

The Magellan and Pioneer Venus Orbiter radiometric tracking data sets have been combined to produce a 60th degree and order spherical harmonic gravity field. The Magellan data include the high-precision X-band gravity tracking from September 1992 to May 1993 and post-aerobraking data up to January 5, 1994. Gravity models are presented from the application of Kaula's power rule for Venus and an alternative a priori method using surface accelerations. Results are given as vertical gravity acceleration at the reference surface, geoid, vertical Bouguer, and vertical isostatic maps with errors for the vertical gravity and geoid maps included. Correlation of the gravity with topography for the different models is also discussed.

Venus gravity and topography: 60th degree and order model

We have combined the most recent Pioneer Venus Orbiter (PVO) and Magellan (MGN) data with the earlier 1978–1982 PVO data set to obtain a new 60th degree and order spherical harmonic gravity model and a 120th degree and order spherical harmonic topography model. Free‐air gravity maps are shown over regions where the most marked improvement has been obtained (Ishtar‐Terra, Alpha, Bell and Artemis). Gravity versus topography relationships are presented as correlations per degree and axes orientation.